IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 __________________________________________________________________________________________ Volume: 03 Issue: 04 | Apr-2014, Available @ http://www.ijret.org 875 VIBRATION ANALYSIS OF LINE CONTINUUM WITH NEW MATRICES OF ELASTIC AND INERTIA STIFFNESS J.C. Ezeh 1 , O.M. Ibearugbulem 2 , C.N. Okoli 3 1 Civil Engineering Department, Federal University of Technology, Owerri, Nigeria 2 Civil Engineering Department, Federal University of Technology, Owerri, Nigeria 3 Civil Engineering Department, Federal University of Technology, Owerri, Nigeria Abstract Vibration analysis of line continuum with new matrices of elastic and inertia stiffness is introduced in this research. The matrices were developed using Ritz method and assumed six term Taylor’s series shape function. Two deformable nodes were introduced at the centre and at the ends of line continuum which brings the number of deformable node to six. The six term Taylor’s series shape function assumed was substituted into strain energy equation and into inertia work (Kinetic energy) equation. Their resulting functional were minimized, resulting in 6 x 6 elastic stiffness matrix and 6 x 6 inertia stiffness matrix respectively, for vibration analysis. The two matrices were employed, as well as the traditional 4 x 4 matrices in classical free vibration analysis of four line continua with different boundary condition. The results from the new 6x6 matrices of elastic and inertia stiffness were very close to exact results, with average percentage difference of 0.212425421% from exact solution. Whereas those from the traditional 4 x 4 matrices and 5 x 5 matrices differed from exact results with average percentage difference of 14.72352281% and 0.275% respectively. Thus the newly developed 6 x 6 matrices of elastic and inertia stiffness are suitable for classical free vibration analysis of line continua Keywords: 6x6 stiffness system; vibration; inertia; line continuum; variational principle; deformable node; shape function; classical; numerical; analysis; beam ---------------------------------------------------------------------***--------------------------------------------------------------------- 1. INTRODUCTION The increase demand for taller structures with high-energy sources (generator), free of foundation vibration and cross winds etc, which create intense vibration excitation problems requires careful analysis and design to avoid resonance or an undesirable dynamic performance. However, the classical approach to line continuum, to obtain exact result, requires formulating the governing differential equation and obtaining the analytical solution. As Moon-Young et al. (2003) pointed out, this analytical method, however, is sometimes inefficient because analytical operations in solving a system of simultaneous ordinary differential equations with many variables maybe too complex. Also Bhavikatti S.S. (2005) pointed out that if structure consists of more than one material, it is difficult to use classical method, but finite element can be used without any difficulty. Moreover, as observed by Ibearugbulem et al (2013), the traditional 4x4 stiffness matrix and its load vector cannot classically analyze flexural line continua except using them numerically (more than one element in one analysis). This difficulty in using the traditional 4 x 4 classical approach is evident in the work of Iyengar (1988), Chopra (1995) and Yoo and Lee (2011). Ibearugbulem et al (2013) developed 5 x 5 stiffness matrices capable of classically analyzing stability and dynamic line continuum by introducing one degree of freedom at the mid span of line continuum, but some of their solutions are not exact solution. This research work will generate new 6 x 6 stiffness matrices for vibration analysis of line continuum by introducing two degrees of freedom (rotation and deflection) at the mid span and at the ends of line continuum which brings the number of deformable node to six. The six term Taylor’s series shape function assumed was substituted into strain energy equation and into inertia work (Kinetic energy) equation. Their resulting functional were minimized, resulting in 6 x 6 elastic stiffness matrix and 6 x 6 inertia stiffness matrix respectively, for classical free vibration analysis. 2. GOVERNING EQUATION The line continuum governing equation is: d 4 w dx 4 − Mω 2 w EI =0 (1) The assumed six term Taylor’s series shape function w(x) = a 0 +a 1 x+a 2 x 2 +a 3 x 3 +a 4 x 4 +a 5 x 5 2